Fringe projector profilometry (FPP) is an important three-dimensional (3D) measurement technique, especially when high precision and speed are required. Thus, theoretical interrogation is critical to provide deep understanding and possible improvement of FPP. By dividing an FPP measurement process into four steps (system calibration, phase measurement, pixel correspondence, and 3D reconstruction), we give theoretical analysis on the entire process except for the extensively studied calibration step. Our study indeed reveals a series of important system properties, to the best of our knowledge, for the first time: (i) in phase measurement, the optimal and worst fringe angles are proven perpendicular and parallel to epipolar line, respectively, and can be considered as system parameters and can be directly made available during traditional calibration, highlighting the significance of the epipolar line; (ii) in correspondence, when two sets of fringes with different fringe orientations are projected, the highest correspondence precision can be achieved with arbitrary orientations as long as these two orientations are perpendicular to each other; (iii) in reconstruction, a higher reconstruction precision is given by the 4-equation methods, while we notice that the 3-equation methods are almost dominatingly used in literature. Based on these theoretical results, we propose a novel FPP measurement method which (i) only projects one set of fringes with optimal fringe angle to explicitly work together with the epipolar line for precise pixel correspondence; (ii) for the first time, the optimal fringe angle is determined directly from the calibration parameters, instead of being measured; (iii) uses 4 equations for precise 3D reconstruction but we can remove one equation which is equivalent to an epipolar line, making it the first algorithm that can use 3-equation solution to achieve 4-equation precision. Our method is efficient (only one set of fringe patterns is required in projection and the speed is doubled in reconstruction), precise (in both pixel correspondence and 3D reconstruction), and convenient (the computable optimal fringe angle and a closed-form 3-equation solution). We also believe that our work is insightful in revealing fundamental FPP properties, provides a more reasonable measurement for practice, and thus is beneficial to further FPP studies.

In comparison to tactile sensors, optical techniques can provide a fast, nondestructive profile/areal surface measurement solution. Nonetheless, high measurement noise, unmeasured points, and outliers are often observed in optical measurement, particularly for structured surfaces. To alleviate their detrimental impacts on the characterization of surface topography as well as the examination of micro/nanoscale geometries, a post processing filtering technique, i.e., the clustering filter, which is essentially an iterative process to find the aggregation center of a cluster of points, is implemented. The clustering filter is particularly useful for noises and outlier suppression for optical measurement of structured surfaces due to its edge-preserving capability. Five surface samples with structured features are measured by an in-house developed dispersive interferometer and a commercial white light interferometer, thereafter the measured surface data are filtered by the clustering filter. Both noise and outliers are suppressed, which not only facilitates the visualization and characterization of surface topography, but also enables the accurate evaluation of local functional geometries.

Film is widely used in optoelectronic and semiconductor industries. The accurate measurement of the film thickness and refractive index, as well as the surface topography of the top and bottom surfaces are necessary to ensure its processing quality. Multiple measurement methods were developed; however, they are limited by the requirements of a known dispersion model and initial values of thickness and refractive index. Further, their systems are rarely compatible with surface topography measurement methods. We propose a constrained nonlinear fitting method to simultaneously measure the thickness and refractive index of film in a simple white-light spectral interferometer. The nonlinear phase extracted by the spectral phase-shifting is fitted with the theoretical nonlinear phase obtained by multiple reflection model. The constraints of nonlinear fitting are obtained by the interferometric signal of vertical scanning, reconstructed by the integration of the white-light spectral signal to avoid local minima. The proposed method does not require a priori knowledge of the dispersion model and initial values of thickness and refractive index, and its system is compatible with the vertical scanning interferometry (VSI) method to reconstruct the surface topography of the top and bottom surfaces of film. Three SiO₂ films with different thicknesses are measured, and the results show that the measured refractive index is within the theoretical value range of wavelength bandwidth and the measured thicknesses are closely aligned with the values provided by the commercial instrument. The measurement repeatability of refractive index reaches 10⁻³. Measurements on a polymer film demonstrate that this method is feasible for measuring the film without a priori information.

Angle-resolved ellipsometry with back focal plane imaging has been found to be of increasing importance in recent industrial sensing by virtue of its rich information provided at various incident and azimuthal angles. To achieve high sensing accuracy, the incident angles of a back focal plane must be accurately calibrated. For this purpose, a simple and robust incident angle calibration method based on full-field Brewster angle fitting is proposed, without expensive tools or complex operations. With this method, a back focal plane image is first captured from boundary reflectance through a high-numerical-aperture objective. By extracting annular data from the image, radius-dependent ellipsometric parameters ( ψ , Δ ) are calculated. At the end, the radii of the back focal plane are mapped to the angle of incidence by using a fitted Brewster angle as the reference. The method is validated by simulation and experiments using a homemade angle-resolved ellipsometer and a commercial spectroscopic ellipsometer. The results show that the proposed method provides a 75% error reduction approximately from generally used methods.

The moving least squares (MLS) and moving total least squares (MTLS) method have been widely used for fitting measurement data. They both can be used to achieve good approximation properties. However, these two methods are susceptible to outliers due to the way of determining local approximate coefficients, which leads to distorted estimation. To reduce the influence of outliers and random errors of all variables without adding small weights or setting the threshold subjectively, we present a robust moving total least squares (RMTLS) method, in which an improved least trimmed squares (ILTS) method is used for obtaining the local approximants of the influence domain. The ILTS method divides the nodes in the influence domain into a certain number of subsamples, achieves the local approximants by the total least squares (TLS) method with compact support weight function and trims the node with the largest orthogonal residual from each subsample respectively. The remain nodes from the subsamples are used to determine the local coefficients. The measurement experiment and numerical simulations are provided to demonstrate the robustness and accuracy of the presented method in comparison with the MLS and MTLS method.